# Why should the $\phi^4$ term necessarily cause scattering while a $x^4$ term in anharmonic oscillator only causes correction of energy levels?

Consider an anharmonic oscillator in quantum mechanics, described by the Hamiltonian $$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2+bx^4.$$ The $$bx^4$$ term doesn't cause scattering. The effect of this term only causes the oscillator eigenstates and energy eigenvalues to be modified.

On the other hand, a theory specified by the Lagrangian density $$\mathscr{L}=\frac{1}{2}(\partial_\mu\phi)^2-\frac{1}{2}m^2\phi^2-\frac{\lambda}{4!}\phi^4$$ with $$m^2>0$$ and $$\lambda>0$$ is interpreted as a self-interacting field of mass $$m$$. Here the term $$\lambda \phi^4$$ causes scattering! But there must again be noninteracting states which should at least be solvable perturbatively and which do not scatter off each other?

Does the first case i.e., $$\phi^4$$ theory fall within the purview of time-dependent perturbation theory and the latter within time-independent perturbation theory?